Archive (winter term 2025/2026)
- January 14, 2026
-
Iosif Petrakis (University of Verona)
title: The "complemented subsets" point of view
abstract:
Bishop and Cheng introduced complemented subsets as positive and strong
counterparts to subsets in their constructive development of the Daniell
approach to measure theory. A complemented subset is a pair of subsets
(A1, A0), where A1 and A0 are
disjoint in a positive and strong sense. While constructively the weak and
strong complements of a subset have a poor algebraic behaviour, the swapped
pair (A0, A1) is a well-behaved notion of a constructive
complement of (A1, A0). In this talk we give an overview
of recent developments within the "complemented subsets" point of view.
The abstract algebraic properties of the complemented powerset define the
notion of a swap algebra, a generalisation of a Boolean algebra, while the
abstract properties of partial, Boolean-valued functions define the notion of a
swap ring, a generalisation of a Boolean ring. An orthocomplemented subspace of
a Hilbert space H is a pair (L1, L0) of orthogonal,
closed subspaces L1 and L0 of H. Orthocomplemented
subspaces correspond to partial projections on H, and provide new models of
constructive quantum logic. Topologies of open complemented subsets constitute
a new approach to constructive point-set topology, while in constructive
computability theory a recursive complemented set is a pair (A1,
A0), where A1 and A0 are recursively
enumerable subsets of N.
Last Change: 2026-03-25, Stefan Hetzl.